If ${ }^{n} C_{12}={ }^{n} C_{8}$,then $n$ is equal to

There are ten boys $B_{1}, B_{2}, \ldots, B_{10}$ and five girls $G_{1}, G_{2}, \ldots, G_{5}$ in a class. The number of ways of forming a group consisting of three boys and three girls,such that $B_{1}$ and $B_{2}$ are not both members of the same group,is

Let $n$ and $k$ be positive integers such that $n \ge \frac{k(k + 1)}{2}$. The number of solutions $(x_1, x_2, ..., x_k)$ where $x_1 \ge 1, x_2 \ge 2, ..., x_k \ge k$ are all integers,satisfying $x_1 + x_2 + ... + x_k = n$,is

The total number of ways of selecting $4$ letters from all the letters of the word $TSEAMCET$ is

$A$ man has $7$ friends. In how many ways can he invite one or more of them for a tea party?

$A$ group consists of $4$ girls and $7$ boys. In how many ways can a team of $5$ members be selected if the team has at least $3$ girls?